JOURNAL
Firm
OF ENVIHONMENTAL
Behavior
Under
ECONOhiICS
AND
of
Economics,
5, 2643
Imperfectly
Enforceable
and Taxes
JON
Department
IvlANAGEhlEiXT
D.
State Unizjersity
Received
HARFORD
of
( 1978)
Pollution
1
New York at Albany,
November
Standards
New York 12222
2, 1976
Assuming
expected
profit
maximization,
the behavior
of the firm under
imperfectly
enforceable
pollution
standards
is examined.
Among
other
results,
it is found
that
cost subsidies
can reduce
the size of violation
and amount
of wastes,
and that the
shape of the expected
penalty
function
determines
the direction
of the firm response
to tighter
standards.
Under
imperfectly
enforceable
pollution
taxes, it is found,
among
other
results,
that the firm’s
actual
level of wastes
is independent
of proportional
changes
in the expected
penalty
for pollution
tax evasion,
and that
the marginal
cost of actual
waste reduction
equals the unit tax on reported
wastes.
Some normative
aspects
of the results
are discussed.
I. INTRODUCTION
In this paper we present simple theoretical models of firm behavior under
imperfectly
enforceable pollution standards and pollution taxes, respectively.
Such models are motivated by the nature of the current and past pollution control efforts in this country and encouraged intellectually by coexisting literatures
on approaches to externalities and the economics of crime. Although the model
presented here has some strong simplifying assumptions, it presents a point of
view distinctly different from most treatments of externalities and allows derivation of some interesting results.
The approach to pollution control adopted by the federal government has by
and large been based upon the use of standards, both ambient and emission
(or effluent). In the case of air pollution, the Clean Air Act of 1970 gives the
Environmental
Protection Agency the power to set emission standards on a wide
variety of sources with the goal of meeting ambient air quality standards. In the
case of water pollution, The Federal Water Pollution Control Act Amendments
of 1972 give broad power to the EPA to set effluent standards on practically all
industrial and municipal sources with the goal of meeting ambient water quality
standards, which, according to legislative intent, are to become increasingly
stringent.
In the case of both air and water pollution standards it has been the case that
these standards have not always been complied with. For support of this understatement with respect to water pollution controls one should consult Kneese’s
of
1 The
New
author
gratefully
acknowledges
York for the researching
and
0095-0696/78/0051-0026$02.00/O
Copyright
All rights
0 1978 by Academic Press, Inc.
of reproduction
in any form reserved.
support
from
writing
of this
the Research
paper.
Foundation
of
the
State
IMPERFECTLY
ENFORCEABLE
STANDARDS
27
paper in Bain [2] and Zwick’s [15] even less charitable view. In fact, major
aspects of the recent water pollution legislation have been aimed at reducing the
costs of enforcing pollution standards by making the EPA’s power greater and
more sharply defined. Both pieces of legislation mentioned have clauses setting
penalties for violations of standards.
It may be that once full monitoring
of pollution sources is effective, the
probability
of successfully violating standards without being caught and punished will be close to zero. However, past experience does not support the view
that perfect enforcement of pollution standards would be of trivial costs in
comparison with the benefits.
Because of the above considerations it is of interest to examine the behavior
of a profit maximizing firm under imperfectly
enforced pollution
standards.
However, it is also of interest to examine the behavior of the firm under imperfectly enforceable pollution taxes. One reason would be the substantial literature that has been devoted to proving that under perfect competition the tax
per unit of pollution equal to marginal damages with no compensation of victims
is the Pareto optimal policy (and that in general virtually all other solutions
cannot attain Pareto optimality).
Baumol and Oates [4] present a general proof
of this proposition and discuss some of the substantial literature that led up to
this conclusion. A second reason for studying the effect on the firm of an imperfectly enforceable pollution tax is that it provides us with a comparison set
of results for the standards case. We find, in fact, that the pollution tax case does
have interesting properties, the main one being a type of separability between
the amount of pollution released and the amount of unreported pollution, or tax
evasion, that occurs.
If there is a second major federal approach to pollution control it would be
that of providing cost subsidies. For firms this is done mostly through the special
form of accelerated depreciation allowed on pollution control capital as provided for in the Tax Reform Act of 1969. Such subsidies, by themselves, would
seem to provide no positive incentive for pollution control. They merely reduce
the cost of doing something which adds nothing to revenues. However, in a world
of imperfectly enforceable controls, we can show that such subsidies reduce the
tendency to violate standards.
Lastly, we will point out some implications of the analysis for the characterization of optimal levels of pollution control. Since firms that violate standards and
evade taxes change the trade-offs in the economy one would expect a change
in the set of efficient resource allocations in comparison with a world of perfectly
costless enforcement or perfectly compliant firms.
In Section II we will examine the expected profit maximizing firm’s response
to imperfectly enforceable pollution standards. In Section III we examine the
firm’s behavior under evadeable pollution taxes. In Section IV we discuss some of
the normative implications of the analysis, and Section V presents some conclusions.
II. THE
FIRM
RESPONSE
TO STANDARDS
Let us consider a firm which produces a good x from which
R(x), The firm may be either a perfect competitor of some
competitor. The costs C of producing the good x depend, not
tity of X, but also on the amount of emitted wastes w. So
it receives revenue
form of imperfect
only on the quanby definition C =
28
JON D. HARFORD
C( X, zu), This cost function is taken to reflect all of the process changes which
would reduce created wastes and any efforts which would “neutralize” wastes
already created. Emitted wastes are those which are created and not neutralized,
and all references to firm wastes (or actual wastes) will mean emitted wastes as
here defined.
We will indicate partial derivatives by subscripts. Thus, marginal output costs
and marginal revenue are C, and R,, respectively. Under usual circumstances
we would expect JR,, - C,] < 0, where the subscripts indicate second partial
derivatives, which we will assume exist for all of the relevant functions unless
noted otherwise. In order for the firm to find it profitable to emit pollution in the
absence of any external constraints it must be true that C, is negative over a
range from zero pollution up to some level ujO. “uJ,,” is the point where C, = 0,
and would be the amount of pollution released in the absence of any controls.
We may assume that C, becomes positive after this point; that is, it would increase costs to release any more pollution. Of course, the firm would never
operate on this portion of its cost curve. Consistent with this description, we
will assume that C,, > 0. If costs were separable in pollution and output we
would have C,, = 0, but it is more reasonable to assume that C,, = C,, < 0,
over the relevant range of the cost function.
Now assume that a particular pollution control board (PCB ) or the EPA
imposes a particular allowable level of released wastes, labeled wS, which is less
than what the firm would have otherwise chosen. The PCB wishes to see this
standard complied with. Therefore it sets penalties in the form of fines (and
perhaps prison terms) and expends resources on activities which create a probability of discovering and convicting standards violators greater than zero but
less than one. For present purposes, the discovery of a violation, any adjudication process, and the receipt of a penalty for the violation will be treated as one
event. For any size of violation, o = (20 - zo,), the firm sees the probability of
discovery and punishment and the level of fine as parameters.
Generally we would expect that any fine f will be a positive function of the
size of violation. It is also plausible that the probability p of detecting a violation
will be a positive function of the size of violation. Thus we would suppose that
f Q and p, are ordinarily greater than zero. We will be assuming risk neutrality
for the firm, so that the expected fine, 6 = pf, is all that matters to the firm. To
facilitate the analysis and reduce the number of terms let us write 6 = F,PG( u),
where F, and P are shift parameters for the general size of fines under standards
and the general level of the probability of detection and punishment of violators,
respectively. Because of their nature, the terms F, and P can be taken to be
equal to one without loss of generality. Thus, G(U) = f (u )p( u), and it is assumed to be continuous and twice differentiable;
and
G’ = f’p + p’j’
G” = f’p
+ 2p’f’
(14
+ p”f,
(lb)
where the primes indicate the order of differentiation.
It is conceivable that there could be a discontinuity
at 0 = 0. This would
require that there be a sudden jump from zero to some positive value in the
probability
of punishment as one goes from no violation to one of arbitrarily
small size and that there be some minimum level of fine. We mention this as
a possibility since such a discontinuity would tend to produce extreme solutions.
IMPERFECTLY
ENFORCEABLE
29
STANDARDS
As a practical matter, however, laws are seldom enforced in such a way as to
produce a significant discontinuity
of this sort. Before going on it should be
noted that both f” and p” may be negative and still have G” positive.
The structure of the fines is something to be determined by the PCB and
presents an optimization problem. It may be suggested that characterizing the
optimal structure of fines is an interesting problem and quite possibly a very
difficult one in general. The probability of detection as a function of the size of
violation might be either something given in the nature of the production function of detection, or a result of conscious allocation of resources. For a situation
involving different types of crimes, the latter interpretation
is more plausible
(and the concept of crime size less clear) while for crimes of the same type this
writer would lean toward the former interpretation.
In either case, we assume
the firm views these functions as given, and, given risk neutrality, is concerned
only with the product of these functions and its derivatives.
In order to complete our picture, let us assume that the firm receives a subsidy
at a rate b, where b is between zero and one, for the costs of eliminating wastes.
In reality subsidies usually apply only to explicit capital expenditures on pollution control and seldom or never to the costs of all potential ways of reducing
pollution, which would include process changes and non-capital inputs into
waste neutralization.
Indeed, it would be difficult to divide costs between output
and pollution control if costs are not additively separable in the technical sense.
We will ignore this practical difficulty and assume the total subsidy is b (C (x, UJ)
- C( X, w,) ) where w, is taken to be the level of wastes that the firm would
produce in the absence of pollution controls and cost subsidies. Since 2~;~is a
constant, we may define C( x, w,) = C”(X).
Given the previous definitions, expected profits, rI,, is of the following form:
II, = R(z) - C”(q w) + b[C(x, w) - P(z)]
- PF,,G(u).
(2,
The firm is assumed to choose output x and the level of waste 20 so as to
maximize expected profits. The assumption of risk neutrality is not perfectly
general, of course, but even for nonrisk neutral firms risk neutrality is a good
approximation
for relatively small risks, which is probably a fair description of
most risks associated with violating pollution standards.2
The first order necessary conditions for an interior maximum are:
fi, - c, + b(C’, - e,q
= 0
(3s)
- (1 - b)C, - I’F,G’
= 0.
(3b)
Condition
(3a) says that marginal revenue should equal marginal costs less
the subsidized fraction of the difference between the marginal costs of output
at the base level of waste and the marginal costs of output at the chosen level of
s Given
the basic tenet of portfolio
theory
that one can evaluate
the. risk of any asset only
in the context
of its relationship
to the array of all assets, it is difficult
to see how one would
make rigorous
sense out of the idea of either
risk averseness
or risk preference
for a firm that
may be owned
by thousands
of stockholders
with differing
portfolios
and preferences
for risk.
The present
assumption
can at least be defended
on the grounds
that
in the long
run
(assuming
survival
of the firm and some version
of the law of large numbers)
the risk neutral
firm will have had greater
average
profits
than either
the risk averse
or risk preferring
firm.
To put it another
way,
if all profits
were
reinvested
for all firms
at some constant
rate of
return,
then the risk neutral
firms would
eventually
be the largest.
JON
30
D.
HARFORD
wastes. If the cost function is separable in x and 20, the last term is zero and we
get the conventional answer that marginal revenue should equal marginal cost.
If C,, < 0, we would expect (C, - C,O) > 0, and therefore, that marginal
revenue would be less than marginal costs. With such a cost function and subsidy one ends up subsidizing output as well as pollution control. This is no doubt
part of the reason why most actual cost subsidies are for types of pollution control equipment that have no obvious connection to the production of output.
This, of course, means that pollution control is not achieved in a least cost
manner from society’s viewpoint.
Condition
(3b) tells us that the unsubsidized fraction of marginal costs of
reducing wastes should equal the increase in the expected monetary value of the
fine with a one unit increase in actual waste. Clearly it may often be the case
that equality in (3b) does not hold. For example, if both the fine and the
probability of receiving a penalty are fixed and independent of the violation size,
then G’ = 0, and C,, = 0 will appear to be the maximizing condition, which is
what it would be in the absence of standards. However, if the expected fine is
high enough one will get the other “comer” solution where w = u;*.
The two possibilities for the case just mentioned are illustrated in Fig. 1. We
have drawn costs as a function of wastes while holding output constant. For
simplicity we have assumed a zero subsidy in this and following figures. Two
different fixed sizes of expected fine are indicated by G1 and Ga, The level of
costs under complete compliance is indicated by Ca, and under no effort at
compliance the costs are Ci. Since Ci + Gi < Ca, the firm makes no effort at
pollution
control under expected fine G,. Under expected fine GB, the firm
chooses zero violation since C1 + Gz > Cz.
In general, the possibilities for corner solutions depend upon the relative
heights and shapes of the cost and expected fine functions, as well as the levels
of w, and b. Figure 2 shows a case where an interior solution, characterized by
w0 > w* > ws, exists. The full compliance Ievel of costs OMs is greater than the
costs at the minimum point of the C + GS curve where wastes are w* and costs
are at height OMi. It is due to the increasing nature of the expected fine and the
decreasing marginal costs of waste prevention that 2~‘~ is to the left of wO.
FIG.
1. Choice
of violation
size
with
constant
expected
fine.
IMPERFECTLY
ENFORCEABLE
STANDARDS
31
C,GI
FIG.
2. Choice
of violation
size with
a constant
slope
expected
fine
function.
Careful perusal of variations on Fig. 2 indicates that as long as G” Z 0, a
(further)
requirement for a positive violation level is that
given a zero subsidy and the other assumptions on the cost function. Since the
marginal costs of reducing wastes decrease as wastes increase, while the slope
of the expected fine function is here assumed to be non-decreasing, we must
have marginal costs of waste reduction at the allowed level of wastes greater
than the rate of increase in the expected penalty with respect to the first unit of
illegal wastes in order to have a positive violation.
To gain further insight into the possibilities of corner solutions we shall now
consider the second order conditions for a maximum. These conditions are that
the determinant of the Hessian matrix,
’ (R,,
-
(1 -
b)C,,
-
H, =
-(l
- b)Czw
bC,Z”)
-(l
-
- (1 - b)C,,
b)C,,
- PF,G”
> 0,
(4)
and that the terms on the diagonal be negative. This expression has the common
implication that the cross partials of the cost function should not be too large
in comparison with the repeated partials in 20 and x. A further implication is
that G” cannot be too strongly negative or the second order conditions will
not hold. One may note that the linear expected fine function in Fig. 2 implies
that its second derivative is zero, and thus it is consistent with satisfaction of the
second order conditions. The satisfaction of the second order conditions ensures
us that there is at most one interior maximum, which will be the global maximum
if there is no discontinuity
in the expected fine function at the zero violation
level.
We may now list and discuss a number of comparative static results showing
rates of change of controlled variables with respect to parameter changes. In
the following results we have taken advantage of the normalization P = F, = 1.
at0
a20
aP = dF,
=-
[Rzz -
(1 - b)C,, - bC,,“IG”
H,
< o
(54
JON D. HARFORD
32
al0
-=
at0,
-[Rzz
-
(1 - O)C,, - bC,,“]C”
> o
<,
as
(;I’, 2 0.
H,
at0
-CR,,
-=
ab
-
(1 - b)C,, - bC.c,“]C, + (1 - b)C,,(C,
_________
____--__-HS
(5h)
- Cz”) < o (5c)
__-
ax
ax
(1 - b)C,,G’
-= -E---------<o
aP
dF,
HS
(54
ax
-aw,
(5e)
- (1 - b)C,,G”
-20
H,
as C” 2 0, assuming C,, < 0.
ax
-iii;
=
(‘2, - Cz”)[(l
- b)Cw
+ G”‘] HS
(1 - b)C,Cw
> o
< *
(50
These results are based on the assumption that both the first and second order
conditions for a maximum are satisfied. On this basis the relationships in (5a)
tell us that wastes created and released will decrease with increases in the level
of fines or increases in the probability
of detection at all sizes of violation. In
other words, a general increase in expected fines decreases the actual level of
wastes and the size of violation which the firm chooses. We also find from (5b)
that an increase in the cost subsidy will reduce the size of any standards violation, a result due basically to the fact that the relative cost of complying with
the standard has been reduced. This provides a rationale for the commonly
used cost subsidies, which economists have tended to ignore or presume to be
useless in situations where none of the benefits of reducing pollution are internal
to the firm.
On the output side we find that increased fines and probability of detection
tend to reduce the expected profit maximizing level of output to the extent that
marginal costs of output are increased by reductions in the emitted wastes. Thus,
without separability we would tend to find that limitations on the amount of
wastes would tend to reduce the profit maximizing level of output for the firm.
Interestingly enough, the direction of change in waste and output levels with
respect to a change in the allowable wastes under the standard is dependent
upon G”, the second’derivative
of the expected fine function with respect to the
size of the violation. If the slope of G is increasing, then waste and output will
move in the same direction as allowable wastes under the standard. This is the
more intuitive result. One may conjecture that it is likely though not certain that
this would be the most relevant case. If G” > 0, it can be plausibly argued from
the relevant expressions that the increase in wastes will be less than the allowable
wastes under the standard, i.e., that the size of violation will decrease with a
loosening of the standard.
However, there is nothing in the second order conditions which definitely
rules out the case where the slope of the expected fine function is decreasing.
In this case we find that making the standard marginally stricter will increase
the actual amount of waste released by the firm. To give the reader a better feel
for the situation we refer to Fig. 2, which illustrates what occurs in the border
IMPERFECTLY
ENFORCEABLE
STANDARDS
33
line case where the slope of the expected penalty function is constant. The shift
in the wastes standard from wB to wg8 has the effect of shifting upward the curve
representing the sum of costs plus the expected penalty to C + Ga. Thus costs
at w* increase from a height of OM, to ONr. However, due to the parallel
nature of the shift, ON1 now represents the new minimum of the sum of costs
plus expected penalty. In other words, there is no change in actual wastes
emitted. The size of the violation increases in the same magnitude as the reduction in the allowed level of wastes.
Finally, we note that the change in output with respect to a change in the
cost subsidy rate is in general ambiguous in sign. This is due to the existence of
two counteracting
effects. The subsidy reduces the cost of pollution control
which allows more output for the same pollution control efforts and total costs.
However, because the subsidy encourages less pollution it indirectly increases
the marginal costs of output. Which effect dominates depends upon the size of
practically all of our parameters and the first and second partial derivatives of
the cost function. In the special case of separability between output and wastes
in the cost function, we can assert that the change in output with respect to a
change in the subsidy rate will be zero.
All of these comparative static results are under the assumption that the first
order conditions are satisfied with equality and the second order conditions
hold. If, on the other hand, we do not have an interior solution, we would still
expect that small changes in the parameter values would have either no effect,
or effect waste and output levels in the direction indicated by the comparative
statics results.
This expectation tends to hold up even with respect to variations in w8 in the
case where complying with the standards maximizes expected profits. One might
be tempted to expect that permitted and actual wastes would move in the same
direction regardless of the shape of the expected fine function. That this may
not be the case is illustrated within Fig. 1. When the allowable level of wastes
is decreased from w, to zu,, under the assumption that the expected fine is a constant Gz we find that overall expected costs are now lowest at the level of w,
wastes, i.e., costs at full compliance with the stricter standard Cs are greater
than (C, + Gz). Thus, in this case, increasing the strictness of the standard
increases the actual level of wastes from wS to w,. To put it another way, the
firm may go from full compliance to totally ignoring the standard as it increases
in strictness if the expected fine does not increase rapidly enough with the size
of violation. For the special case illustrated by Fig. 1, Anderson [l] preceded
us in discovering this result.
All of the previous results are about individual firms and, therefore, one needs
to know something about industry structure in order to discuss effects of pollution standards on aggregate of firms. For firms operating in an environment
where expected long run economic profits should be zero, average costs become
important for determining the amount of wastes which the industry will generate
since average costs determine the size of the industry in terms of output. Thus,
even in the seemingly perverse case where tightening standards increases the
firm’s waste creation, we would expect (and Figs, 1 and 2 confirm) that the
firm’s average costs would increase. This would tend to reduce the size of a competitive industry and work in the opposite direction from the “counterproductive”
effect on the individual firm.
34
JON
D.
HARFORD
In the same vein, an increase in the cost subsidy rate for pollution control
would be expected to reduce the average costs of output for the firm even though
the marginal effect is uncertain, Thus the size of the industry would be larger
with the subsidy, ceterus paribus, even though each individual firm may produce
less pollution. It is possible that the larger industry size would increase pollution
relatively more than any reduction in the pollution per unit of output that the
cost subsidy might induce. The overall effect will depend upon a number of
factors including the elasticity of the market demand for the output.
III.
EVADEABLE
POLLUTION
TAXES
It is of interest to compare the kind of results we obtain with pollution standards with those we would get in the case where a tax per unit of waste emitted
is imposed. To make the comparison reasonable, it is assumed that the pollution
tax is evadeable. This may be interpreted as meaning that actual wastes released,
w, are greater than the reported wastes, 20r, where the tax is directly on reported
wastes only. The firm’s costs C are a function of ouput x and the actual wastes
released w in the same manner assumed in the case of standards. (We no longer
assume any cost subsidy.) All of the remarks about the signs of the derivatives
of the cost function still apply.
The unit tax t is applied to w, so that total pollution tax paid equals t times
w,.. In order to prevent the firm from evading all taxes by reporting zero released
wastes, the pollution taxing agency imposes penalties for evasion of pollution
taxes. The size of the evasion will be measured by 2) = (w - w,.), where we
re-use the letter 2) to reflect the size of violation in this different case. There is
some justification for this procedure. In the cases of standards, the reported
wastes would always be w, since there is no incentive to have actual or reported
wastes less than ws, and reported wastes above w, would be asking for a penalty.
So even in the standards case the violation size v could be interpreted as the
difference between actual and reported wastes.
Again we will use the function G(v) = pf to reflect the shape of the expected
fine as a function of the violation size where p and f have the same meaning as
before. The shift parameter of the probability of detection and punishment will
again be labeled P, which will be taken to be equal to one. We shall distinguish
between two components of what we shall call the expected penalty function.
The first component consists of a fine shift parameter F, also taken to be equal
to one, multiplied by PG( V) to reflect the expected fine size. The second component will be the tax that will be collected on unreported
but discovered
wastes. Thus, in addition to any expected fine and the tax on reported wastes,
there is an expected tax on unreported wastes of Pput. To reduce the number of
terms in later expressions we will define B = pv. We note that
B’ = p + p’v
B”
=
Q’
+
(6a)
p”v.
If we again use R(x) to denote total revenue as a function
write expected profits as
n, = R(z) - C(z, w) - fw, - FPG(v)
of output,
- PB(v)t.
we may
Vi
The firm now has three control variables, 3c,w, and wT, with which it attempts
IMPERFECTLY
ENFORCEABLE
STANDARDS
35
to maximize expected profits. Accordingly, we will have three first order necessary conditions for a maximum, which are:
Ii, - c, = 0,
- c, - FPG’-PB’t=
-t
--C,-G’-B’t=O,
+ FPG’ + PB’t = -t
@aI
(Sb)
+ G’ + B’t = 0.
(Xc)
Condition (8a) informs us that marginal revenue should equal marginal cost,
the same type of result one would get if there were no pollution tax (although
the level of marginal costs will be affected by the actual level of pollution control). Condition (Sb) indicates that the marginal cost of reducing actual waste
should be equal to the marginal increase in the expected penalty (expected fine
plus additional tax) with a unit increase in violation size. Condition (SC) tells
us that the increase in expected penalty from a unit decrease in reported wastes
should equal the unit tax on reported wastes.
If these conditions hold, it is obvious but interesting the -C, = t. In other
words, the marginal cost of actual pollution reduction by the firm will equal the
unit tax on report& pollution. This implies that the actual waste level of the
firm does not directly depend upon the size of our shift parameters for the fine
or the probability
of discovery of the violation. Furthermore,
if the expected
punishment levels are generally so high that the firm maximizes expected profits
by having actual and reported wastes the same [ ( G’ -+ Et) > t at all positive u
would cause this situation], then it would still clearly be true that the marginal
costs of waste reduction would equal the unit tax on reported wastes, since reported and actual wastes are the same.
At the other extreme of corner solutions, however, this result would not hold.
If expected punishment costs are sufficiently low and increase so slowly that the
unit tax on reported wastes is everywhere greater than the increase in expected
penalty with respect to violation size, one would get a case where reported
wastes are equal to zero, and the marginal costs of actual waste reduction would
be equal to the smaller magnitude of the increase in expected penalty costs.
Practically speaking, corner solutions where reported wastes are zero are
unlikely. It would be irrational to set penalties so low that no pollution tax at
all was collected. Moreover, if it is obvious that every firm generates some
wastes, reporting zero wastes would be a clear signal of a violation.
Deriving the Hessian matrix of second order derivatives we can characterize
the second order conditions for a maximum in terms of restrictions on the signs
of its determinant
and principal minors. We have the restrictions that the
determinant
/ (L
H, =
- C*,)
- C’,,
0
-c,,
0
(-C,.,-A)
A
A
-A
I
<o,
(9,
where A = G” + B”t, and that all the diagonal elements must be negative,
while all second order minors should be positive. These restrictions imply that
the expression represented by A must be positive for an interior maximum to be
unique, and a global maximum if there is no discontinuity
in the expected
36
JON D. HARFORD
penalty function at the zero violation level. Thus, the expected penalty function
must have an increasing slope to satisfy the second order conditions and provide
the possibility of an interior solution in the pollution tax case, whereas the expected fine function did not have to have a positive curvature in order for an
interior solution to exist. It is still true, however, that the expected fine function,
by itself, does not have to have a positive curvature if the function I? has a
sufficiently strong positive curvature.
Although we have a larger matrix, the comparative statics analysis turns out
to be simpler in some ways than in the standards case. Equating the shift
parameters to one we have the following results:
&U
--=-=
aw
aP
aF
0
ax
ax
-- =-=
(lob)
0
aP
aF
al0
--=
at
ax
- FL, - CzzM < o
Ht
--
WC)
- C,,B
=-<o
at
Ht
atoT
--=-- ~‘[(Rzz - czxwm + cm21
aF
a2u,
-=
aP
akb
--
at
au
-=
at
-->o
Ht
=
(G’ + B/t)
G’
-[(R,,
a20, , o
(3aF
- C,,>Cmo + Czw21(1 - B’)
dw
__-I___-__+;<o
Ht
aw
--__
at
alu,
at
>o
(104
Cl@>
uw
md
UW
First of all, quite consistently with previous statements, we find that the
actual level of wastes released is independent of the level of fines or probability
of detection and punishment. Small changes in I’ and F have no effect on w.
Furthermore,
small changes in P and F have no effect on the firm’s level of
output. This follows quite intuitively; if the optimal level of actual waste does
not change, then costs as a function of output would not change, and, therefore,
marginal revenue would equal marginal costs at the same output level both
before and after any shift in fines or the probability of punishment.
We find that actual wastes decline as the tax on reported wastes is increased.
This occurs because of the connection of both actual and reported wastes to the
rate of change in the expected penalty with respect to violation size. If costs are
not separable in output and wastes, we find that output is a negative function
of the tax on wastes. If costs are separable, then the firm’s output will not be
directly affected by the tax on reported wastes.
Result ( 1Of) indicates that increases in the probability of capture at all sizes
IMPERFECTLY
ENFORCEABLE
STANDARDS
37
of violation will increase reported wastes. Since actual wastes are not affected
by such a parameter shift, this implies a reduction in violation size at the same
absolute rate. Using this result and explicitly writing out and simplifying ( lOf),
one can derive the following relationship between the elasticity of violation size
with respect to the general probability
of detection, and the elasticity of the
slope of the expected penalty function.
dv
P
-1
z
P
_ E epu = -- 3 --v
( Z’ > v
eaz
(->
dP
(11)
where 2 = G’ + B’t, and use is made of the normalization P = 1.
This relationship says that the elasticity of violation size (in absolute value)
with respect to a proportional shift in the probability of punishment is inversely
r related to the relative positive curvature of expected punishment as a function of
violation size. To say it another way, if the rate of increase in the increase in
expected punishment with violation size is relatively small, then a proportional
increase in the probability of being fined at all levels of punishment will elicit
a relatively large decrease in violation size.
The rate of change in reported wastes with respect to a change in the tax on
reported wastes may be broken up into two parts: A term exactly equal to the
effect on actual wastes, and a term which reflects the direct tax and penalty
effects of a tax shift. It has already been determined that an increase in the
pollution tax reduces actual wastes, and accordingly this component reduces the
reported wastes to the same extent. The other term in (log) will be negative if
(l-73’) is positive. An examination of the first order condition (8~) indicates that
this must be true if the slope of the expected fine function is positive, which we
assume to be the case. Since both terms are negative, reported wastes will
decline to a greater extent than actual wastes when the pollution tax is increased. This, as (10h) indicates, means that an increase in the pollution tax
increases the size of the violation that the firm chooses. As one might expect,
raising the tax rate causes more evasion and presumably a more difficult enforcement problem.
Given the insensitivity of actual wastes to the general level of fines and the
probability of punishment, one might be tempted to conclude that their exact
magnitude is of little or no consequence over a considerable range. This idea
has a rough validity for all firms in the short run, and firms in noncompetitive
industries in the long run. It is definitely not true in the long run for those firms
in industries where expected long run economic profits tend toward zero. In
these industries it is the long run average costs which determine the industry size
in terms of output. A lower level of enforcement of payment of pollution taxes
may not change the actual level of wastes emitted by any firm, but the implied
lower level of some combination of pollution taxes, and penalty payments will
imply lower average costs and more firms in the industry. In this regard it
should be noted that, ceterus paribus, firms with the same expected fine function
will have lower average costs under the imposition of standards than under the
imposition of pollution taxes. It is roughly on this point, as Buchanan and Tullock
[7] explain, that standards fail to produce a Pareto optimal (or pollution tax
equivalent) resolution to a pollution externality under perfect competition.
This model may not be irrelevant to the current situation in the area of water
38
JON
D.
HARFORD
pollution control. The recent federal legislation states that municipalities
must
charge fees to firms for accepting their wastes, fees which must be designed to
cover the costs of treating those wastes. If the firm were not legally allowed to
channel any wastes directly into public waters, then, with proper interpretation,
this model might almost directly apply. The fee for municipal acceptance of
firms’ effluent would correspond to the tax on reported, in this case delivered,
wastes. Violations might take the form of illegal dumping of wastes in the river,
or channeling more wastes to the municipal treatment plant than the firm pays
for in effluent fees. (More complicated and subtle versions of the same problem
might involve the firm reneging on pretreatment
requirements
or negotiating
reductions in other types of taxes (such as the local property tax) to compensate
for the newly instituted effluent fees.) Even if the firm is allowed to release a
fixed level of wastes, as, directly into public waters, we may simply define the
size of violation as o0 = [w - ( zus + We) 1, and the previous results are not 1
qualitatively altered, although chosen values of the control variables would differ.
If one modifies the definition of the violation size as just suggested, one gets
the adsditional results that aw,/a~~ = -1, and au;/au;, = 0. The former result
says that there would be a one to one reduction in reported waste (subject to
effluent fees by the municipality)
with any increase in the allowable standard
of wastes disposable directly into public waters. Put another way this result is
quite striking. It tells us that making the standard stricter would not increase
the violation size, but simply cause an increase in the amount of wastes going
to the treatment plant and paying the effluent fee. The latter result indicates that
changing the standard would have no effect on the actual amount of wastes
created, which is in contrast to the pure standards case, but quite in spirit with
our results in the pollution tax case.
Inevitably one runs into difficulty in trying to fit reality to a model. Relevant
here is the fact that legislatively described fines are often in terms of days of
violation rather than being geared to the quantity of pollutant. Even so, or perhaps because of this, the actual level of fines is left to the discretion of the
courts within the constraint of maximum and minimum fines.3 This methodology
could produce its own peculiarities. Another complicating factor is the actual
variety and interactability
of real world wastes. And, with at least two, and
perhaps three, levels of government (with varying views of benefits and costs)
participating
in pollution control with regard to the same set of potentially
geographically mobile firms, the actual situation becomes extremely complicated.
IV. NORMATIVE
IMPLICATIONS
The analysis heretofore has been predictive in the sense of simply drawing out
the implications for a firm’s behavior under expected profit maximization. We
will now attempt a discussion of the normative implications of an analysis that
includes the possibility of evading the pollution taxes, which are imposed pres For example,
within
Title
III,
Section
309 of the Federal
Water
Pollution
Control
Act
Amendments
of 1972 it is stated
that any firm or municipality
(or the persons
responsible)
violating
an applicable
pollution
standard
“. . . shall be punished
by a fine of not less than
$2250
nor more
than
$25,000
per day of violation,
or by imprisonment
for not more
than
one year,
or by both.”
Except
for the expectation
that
the courts
will use their
discretion
wisely
in gearing
fines to the actual
damages
caused
by the pollution,
this would
appear
to
be an unsatisfactory
way to relate
fines to the severity
of the violation.
IMPERFECTLY
ENFORCEABLE
STANDARDS
39
sumably to reduce an externality to an efficient level. The discussion will speak
in terms of a pollution tax rather than a standard, since it is the former which
has been most favored by economists for its potential to improve efficiency. We
will use the intuitive, but not completely rigorous, concepts of damages and costs
measured in dollar values. Hopefully
this problem may be treated within a
rigorous general equilibrium
framework
at some future point. At this point,
however, the technical problems with more precise approach appear too complex
to explore.
The results derived indicate that the larger the pollution tax, the greater the
evasion of that tax. This points to increasing enforcement costs as attempts are
made to reduce actual pollution. We therefore have increasing marginal costs
of red’ucing pollution which include additional costs of enforcing pollution taxes
(or standards). This would argue for a reinterpretation
of the rule that pollution
should be eliminated to the point where marginal damages are equal to the
marginal costs of treatment or prevention. Under present assumptions we should
have marginal damages equal to the marginal costs of treatment or prevention
plus the marginal costs of enforcing the treatment or prevention.
This point can be argued somewhat more rigorously by a modification
of
Becker’s [5] approach to the optimal level of crime and law enforcement, which
is to minimize the sum of the damage caused by crime plus the costs of capturing and punishing criminals. In this context we wish to minimize the sum of the
damages caused by pollution plus the costs of treating or preventing pollution
plus the costs of enforcing the taxes or other instruments by which pollution
control is encouraged.
Let us define the value of the damage done to society by an aggregate level
of waste W as D(W). On the basis of previous analysis we would expect aggregate wastes to be a decreasing function of the unit pollution tax t. It is also consistent with previous results to assert that the aggregate level of waste should be
a decreasing function of the general level of fines F and the general probability
of detection and punishment for an evasion of pollution taxes, P. (P and F are
now viewed as control variables of the government rather than as parameters.)
Previous results indicate that the later two factors may ordinarily have no effect
on the actual level of the firm, but, as we have argued before, expected punishment levels will effect the number of firms (and thus aggregate wastes) in a competitive industry via the effects on expected average costs. The costs of removing
or preventing wastes are an increasing function of how much lower wastes are
than than they would be if no efforts were made to incur or prevent wastes.
This base level of wastes will be labeled W,, while the aggregate costs of
neutralizing wastes will be S = S(Y), where Y = ( W, - W) .
Costs of enforcement of the pollution tax, E, will be an increasing function of
the probability of capturing violators, P, and an index of the number of violators,
V. No analysis of the shape of the function relating the probability of detection
and the size of the violatiou will be attempted.
However, the variable V is
taken to reflect some index of the number and size of violations, which would
in general depend upon the ease of detecting violations of different sizes. The
level of the index of violations will itself be a negative function of the probability
of detection and punishment, and the general level of fines F, but it should be a
positive function of the height of the unit tax t on reported wastes. Thus, we
have E = E[P, V(P, F, t)].
JON D. HARFORD
40
Lastly, consistent with Becker, it will be assumed that there are social costs
of punishment. These social costs are assumed to be a linear function of the
general probability of punishment times the general magnitude of punishment.
Specifically, total punishment
costs are hPF, where h is a parameter which
reflects general population size and other factors. Again, for simplicity, we avoid
consideration of the structure of fines over the violation size. Given the fact that
punishment in this context is likely to be a fine, the assumption that there are
non-negligible
social costs to punishment may not be very appealing to some.
Harris [9] suggests another rationale for social costs of punishment:
that of
wrongful punishment. Occasionally, the innocent may be fined and this can be
looked upon as causing social costs. On a more general note Stigler [12] points to
the idea of marginal deterrence rather than social costs of punishment as the
critical factor in the determination
the appropriate level of punishment for any
crime. This may be interpreted as saying that the higher the punishment for a
given type of crime the greater the occurrence of other types of crimes with
their attendant social costs. Further discussion of how, and how well, these
rationales serve in arguing for social costs of punishment is not warranted here,
except to say that Stigler’s line of thought quickly leads us back to the tricky
problem of determining
an optimal structure of punishment.4 For the moment
we will simply work with our simple assumption.
We may now write the sum of pollution damage costs plus the costs of treatment/prevention
plus the costs of enforcement of pollution taxes as
L = D(W)
+ S(Y) + E(P, V) + kPF.
(13
The social goal is to minimize total social costs through the choice of F, P, and
t. The three first order necessary conditions for minimum social costs are:
where the subscripts indicate partial derivatives.
(Dw - Sy)Wt + EvVt
(D,
- X,)Wp
= 0
(l&i)
+ Ep + EvVp + kF = 0
(13b)
(Dw - SY)WF + EvVp + kP = 0,
(13c)
For present purposes it is condition (13a) that is of main interest, and so we
will forego any comment on conditions (13b) and (13c), except to say that
they indicate the usual balancing of marginal damages and marginal costs as
they relate to the variables P and F, respectively. Condition (13a) can be rearranged to read
Dw = St- - (EvV’,)!‘Wt.
(14)
4 If one examines
the concept
of marginal
deterrence
closely
it becomes
evident
that,
unless
there
is some maximum
feasible
costless
punishment,
the need for marginal
deterrence
alone will not always
lead one to a finite punishment
for any crime.
The basic reason
for this
is that one can mathematically
conceive
of infinite
rates of punishment.
In the context
of
this paper,
one might
have an infinite
amount
of fine per unit of violation
and thus preserve
marginal
deterrence
without
having
finite
punishments.
Clearly,
in any practical
sense, there
is a maximum
feasible
costless
punishment.
Assuming
for the moment
that fines have
zero
social
costs, it is clear that any fine cannot
exceed
the economic
worth
of the firm
or individual
fined.
Given
that there
is a maximum
feasible
costless
punishment,
the concept
of
marginal
deterrence
should
enable
one to reasonably
consider
what
an optimal
structure
of
punishments
should
be.
IMPERFECTLY
ENFORCEABLE
STANDARDS
41
Given our assumptions about the various functional relationships involved, the
expression to the right of the minus sign is itself negative. This implies that the
damages caused by one more unit of waste should be greater than the costs of
physically eliminating the unit by the amount of the extra expenditure on enforcement induced by the increase in tax evasion caused by the additional pollution tax required to reduce pollution by the last unit.
Of course, a full assessment of the normative implications of pollution controls
must recognize that the imperfections may be just as great in enforcing nonpollution types of taxes and controls. Imperfections in controls are simply another
set of factors to consider in forming policy toward the existence of pollution
types of externalities, along with market structure, various uncertainties,
and
dynamic considerations.
V. CONCLUSION
This paper has presented models of an expected profit maximizing firm under
imperfectly enforceable pollution standards and under imperfectly enforceable
pollution taxes. We have found that under standards increasing the expected
level of the penalty will reduce the level of wastes released by the firm, but
that increasing the strictness of the standard will only reduce the firm’s wastes if
the slope of the expected fine function with respect to the size of the standards
violation is increasing. We have also found that the use of cost subsidies for
pollution control expenses can serve the useful function of reducing the level of
the firm’s violation of the standard and thus its actual level of wastes.
Under imperfectly enforceable pollution taxes the analysis has established the
neat result that the marginal costs of pollution reduction by the firm will be
equated to the constant rate of the pollution tax as long as the slope of the
expected penalty function is increasing. This implies that on the firm level the
amount of pollution tax evasion is independent of the actual wastes, which are
determined by the pollution tax rate. The level of tax evasion is determined by
equating the increase in expected penalty with respect to a unit reduction in
reported wastes to the decrease in tax paid. Therefore, the general level of fines
and the probability of punishment affect reported wastes but not actual wastes.
Further results indicate that increasing the tax rate on reported wastes will
reduce actual wastes released by the firm, but it will reduce reported wastes
even more, implying an increase in pollution tax evasion.
We have also suggested that, with proper reinterpretation
of the violation
size as the difference between actual wastes and an allowable standard of wastes
plus an amount of (reported)
wastes going to a fee charging treatment plant,
one can apply the pollution tax model to current situation in the area of water
pollution.
Under this reinterpretation,
changes in the allowable standard of
wastes has no effect on wastes leaving the firm, or violation size, but are offset
exactly by the amount of wastes going to the treatment plant.
In considering the implications of these results for policy it has been mentioned that effects of various policies on the expected average costs of firms are
important and may not always work in the same direction as the effect on the
individual firm’s marginal decision. Furthermore,
in Section IV, we suggested
a
reinterpretation
of the intuitive rule of efficiency in pollution control that the
marginal damages of pollution should equal the marginal costs of eliminating
42
JON
D.
HARFORD
pollution. The version which this paper suggests is that the marginal damages
should be equal to the marginal costs of physically eliminating the pollution
plus the additional costs of enforcing any pollution control instrument to the
extra degree required to induce the unit reduction in pollution.
Throughout
this paper we have assumed that the pollution control agency
faces firms which believe that they cannot effect the penalty structure or subsidy
rate of the agency. If the number of different polluters is small, this assumption
may not be valid. Firms may adopt various kinds of strategic behavior and
threats in order to affect the penalty structure. They may threaten to go out of
business, or to move to a region where the agency does not have authority.
Firms might collude to violate pollution control laws simultaneously, thereby
overloading the agency’s ability to enforce its laws with any effectiveness. The
agency may be able to adopt various counterstrategies in this regulatory duopoly
game. The outcome of such a situation is not easily determined, but there are
counterparts to it in many types of regulation. Assuming that it is desirable to
minimize the possibilities of such strategic behavior on the part of firms, it appears preferable to formulate and administer pollution controls at the most
inclusive level of government possible.
This paper raises a number of (other) unsolved problems. One of the more
interesting ones is the development
of the concept of an optimal structure of
penalties. This will likely involve an examination of both the structure of fines
and penalties per se, and the technical and resource allocational nature of the
probability of detecting and punishing violations of different sizes. Another problem would be to develop a more explicit and rigorous model to analyze the
optimal level of pollution, pollution taxes, and enforcement of pollution taxes
than that developed in Section IV of this paper. That analysis might be interpreted as suggesting that one should have a lower pollution tax rate in a case
where there are enforcement problems than when there are not, but such a conclusion is not explicit, and we suspect that it may not always be true. A more
explicit analysis would clarify this and other significant ambiguities.
REFERENCES
1. Robert
J. Anderson,
Jr., Environmental
management
economics,
in “Regional
Environmental
Management:
Selected
Proceedings
of the National
Conference,”
(Coate
and
Bonner,
Eds. ), pp. 201-208,
John Wiley
and Sons, New
York
( 1975).
2. J, S. Bain (Ed.),
“Environmental
Decay,”
Little
Brown
and Company,
New York,
(1973).
3. W. J. Baumol,
On taxation
and the control
of externalities,
Amer.
Econ.
Rev. 62, 307322 ( 1972).
4. W. J. Baumol
and W. E. Oates,
“The
Theory
of Environmental
Policy,”
Prentice-Hall,
Englewood
Cliffs
( 1975).
Becker,
Crime
and punishment:
An economic
approach,
J.
(1968).
J. M. Buchanan,
External
diseconomies,
corrective
taxes,
and
Econ. Reu. 59, 174-177
( 1969).
J. H. Buchanan
and G. Tullock,
Polluters
profits
and political
versus
taxes, Amer.
Econ. Reu. 65, 139-147
( 1975).
M. E. Darby
and E. Karni,
Free competition
and the optimal
Econ. 67-88 ( 1973).
J. R. Harris,
On the economies
of law and order,
J. Polk Econ.
Econ.
6, 169-217
5. G.
Polit.
6.
market
structure,
Amer.
response:
Direct
controls
7.
8.
9.
amount
of fraud,
78, 165-174
J. Law
(1970).
IMPERFECTLY
10.
11.
12.
13.
14.
15.
ENFORCEABLE
STANDARDS
43
S. Rose-Ackerman,
The economics
of corruption,
J. Pub. Econ.
4, 187-203
( 1975).
W. Schulze
and R. C. d’Arge,
The Coase
proposition,
information
constraints,
and longrun equilibrium,
Amer.
Econ. Rev. 64, 763-772 ( 1974).
G. J. Stigler,
The optimum
enforcement
of the laws, J. PO&. Econ. 78, 526-536 ( 1970).
U. S. Congress,
“Clean
Air Amendments
of 1970,”
(PL91-604)
USGPO,
Washington,
D. C. 1970.
U. S. Congress,
“The
Federal
Water
Pollution
Control
Act
Amendments
of 1972,”
(PL92-500)
USGPO,
Washington,
D. C. ( 1972).
D. Zwick,
“Water
Wasteland,”
Bantam,
New
York
( 1972).